scholarly journals Approximating common fixed points of two nonexpansive mappings in Banach spaces

1998 ◽  
Vol 57 (1) ◽  
pp. 117-127 ◽  
Author(s):  
Sachiko Atsushiba ◽  
Wataru Takahashi
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Fréchet Differentiable ◽  
Image Position

Let C be a nonempty closed convex subset of a real Banach space E and let S, T be nonexpansive mappings of C into itself. In this paper, we consider the following iteration procedure of Mann's type for approximating common fixed points of two mappings S and T:where {αn is a sequence in [0,1]. Using some ideas in the nonlinear ergodic theory, we prove that the iterates converge weakly to a common fixed point of the nonexpansive mappings T and S in a uniformly convex Banach space which satisfies Opial's condition or whose norm is Fréchet differentiable.

scholarly journals Common Fixed Points Iteration Processes for a Finite Family of Asymptotically Nonexpansive Mappings

2004 ◽  
Vol 11 (1) ◽  
pp. 83-92
Author(s):  
Jui-Chi Huang
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive ◽  
Iteration Processes ◽  
Suitable Sequence

Abstract Let 𝐸 be a uniformly convex Banach space which satisfies Opial's condition or its dual 𝐸* has the Kadec–Klee property, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑇𝑗 : 𝐶 → 𝐶 an asymptotically nonexpansive mapping for each 𝑗 = 1, 2, . . . , 𝑟. Suppose {𝑥𝑛} is generated iteratively by where 𝑈𝑛(0) = 𝐼, 𝐼 is the identity map and {α 𝑛(𝑗)} is a suitable sequence in [0, 1]. If the set of common fixed points of is nonempty, then weak convergence of {𝑥𝑛} to some is obtained.

Two-step iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings

2015 ◽  
Vol 08 (03) ◽  
pp. 1550060
Author(s):  
Amit Singh ◽  
R. C. Dimri ◽  
Darshana J. Prajapati
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Iterative Approximation ◽  
Convergence Theorems ◽  
Strong And Weak Convergence

In this paper, we study an iterative approximation of common fixed points of two nonself asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in a uniformly convex Banach space.

scholarly journals Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space

2007 ◽  
Vol 38 (1) ◽  
pp. 85-92 ◽  
Author(s):  
G. S. Saluja
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Convex Subset ◽  
Convergence Theorem ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Strong Convergence Theorem

In this paper, we study strong convergence of common fixed points of two asymptotically quasi-nonexpansive mappings and prove that if $K$ is a nonempty closed convex subset of a real Banach space $E$ and let $ S, T\colon K\to K $ be two asymptotically quasi-nonexpansive mappings with sequences $ \{u_n\}$, $\{v_n\}\subset [0,\infty) $ such that $ \sum_{n=1}^{\infty}u_n

scholarly journals Approximating common fixed points of families of quasi-nonexpansive mappings

1995 ◽  
Vol 18 (2) ◽  
pp. 287-292 ◽  
Author(s):  
M. K. Ghosh ◽  
Lokenath Debnath
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Special Cases

This paper deals with a family of quasi-nonexpansive mappings in a uniformly convex Banach space, and the convergence of iterates generated by this family. A fixed point theorem for two quasi-nonexpansive mappings is then proved. This theorem is then extended for a finite family of quasinonexpansive mappings. It is shown that Ishikawa's [1] result follows as special cases of results proved in this paper.

scholarly journals Approximating Common Fixed Points of Nonexpansive Mappings in Banach Spaces

2006 ◽  
Vol 13 (3) ◽  
pp. 529-537
Author(s):  
Naseer Shahzad ◽  
Reem Al-Dubiban
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence

Abstract Let 𝐾 be a nonempty closed convex subset of a real uniformly convex Banach space 𝐸 and 𝑆, 𝑇 : 𝐾 → 𝐾 two nonexpansive mappings such that 𝐹(𝑆) ∩ 𝐹(𝑇) := {𝑥 ∈ 𝐾 : 𝑆𝑥 = 𝑇𝑥 = 𝑥} ≠ ø. Suppose {𝑥𝑛} is generated iteratively by 𝑥1 ∈ 𝐾, 𝑥𝑛+1 = (1 – α 𝑛)𝑥𝑛 + α 𝑛𝑆[(1 – β 𝑛)𝑥𝑛 + β 𝑛𝑇𝑥𝑛], 𝑛 ≥ 1, where {α 𝑛}, {β 𝑛} are real sequences in [0, 1]. In this paper, we discuss the weak and strong convergence of {𝑥𝑛} to some 𝑥* ∈ 𝐹(𝑆) ∩ 𝐹(𝑇).

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

scholarly journals Weak convergence theorem for Passty type asymptotically nonexpansive mappings

1999 ◽  
Vol 22 (1) ◽  
pp. 217-220
Author(s):  
B. K. Sharma ◽  
B. S. Thakur ◽  
Y. J. Cho
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Weak Convergence Theorem ◽  
Fréchet Differentiable

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

scholarly journals Approximating fixed points by ishikawa iterates

1989 ◽  
Vol 40 (1) ◽  
pp. 113-117 ◽  
Author(s):  
M. Maiti ◽  
M.K. Ghosh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex

In a uniformly convex Banach space the convergence of Ishikawa iterates to a fixed point is discussed for nonexpansive and generalised nonexpansive mappings.

scholarly journals Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

2001 ◽  
Vol 27 (11) ◽  
pp. 653-662 ◽  
Author(s):  
Jui-Chi Huang
Keyword(s):  
Banach Space ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Bounded Sequences

LetEbe a uniformly convex Banach space,Ca nonempty closed convex subset ofE. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by{Tj:C→C}j=1ras follows:Un(j)=an(j)I+bn(j)TjnUn(j−1)+cn(j)un(j),j=1,2,…,r,x1∈C,xn+1=an(r)xn+bn(r)TrnUn(r−1)xn+cn(r)un(r),n≥1, whereUn(0):=I,Ithe identity map; and{un(j)}are bounded sequences inC; and{an(j)},{bn(j)}, and{cn(j)}are suitable sequences in[0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.

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